# statefbk.py - tools for state feedback control # # Initial authors: Richard M. Murray, Roberto Bucher # Creation date: 31 May 2010 """Routines for designing state space controllers.""" import warnings import numpy as np import scipy as sp from . import statesp from .config import _process_legacy_keyword from .exception import ControlArgument, ControlSlycot from .iosys import _process_indices, _process_labels, isctime, isdtime from .lti import LTI from .mateqn import care, dare from .nlsys import NonlinearIOSystem, interconnect from .statesp import StateSpace, _ssmatrix, ss # Make sure we have access to the right Slycot routines try: from slycot import sb03md57 # wrap without the deprecation warning def sb03md(n, C, A, U, dico, job='X',fact='N',trana='N',ldwork=None): ret = sb03md57(A, U, C, dico, job, fact, trana, ldwork) return ret[2:] except ImportError: try: from slycot import sb03md except ImportError: sb03md = None try: from slycot import sb03od except ImportError: sb03od = None __all__ = ['ctrb', 'obsv', 'gram', 'place', 'place_varga', 'lqr', 'dlqr', 'acker', 'place_acker', 'create_statefbk_iosystem'] # Pole placement def place(A, B, p): """Place closed loop eigenvalues. K = place(A, B, p) Parameters ---------- A : 2D array_like Dynamics matrix. B : 2D array_like Input matrix. p : 1D array_like Desired eigenvalue locations. Returns ------- K : 2D array Gain such that A - B K has eigenvalues given in p. Notes ----- This is a wrapper function for `scipy.signal.place_poles`, which implements the Tits and Yang algorithm [1]_. It will handle SISO, MISO, and MIMO systems. If you want more control over the algorithm, use `scipy.signal.place_poles` directly. Limitations: The algorithm will not place poles at the same location more than rank(B) times. References ---------- .. [1] A.L. Tits and Y. Yang, "Globally convergent algorithms for robust pole assignment by state feedback, IEEE Transactions on Automatic Control, Vol. 41, pp. 1432-1452, 1996. Examples -------- >>> A = [[-1, -1], [0, 1]] >>> B = [[0], [1]] >>> K = ct.place(A, B, [-2, -5]) See Also -------- place_acker, place_varga """ from scipy.signal import place_poles # Convert the system inputs to NumPy arrays A_mat = _ssmatrix(A, square=True, name="A") B_mat = _ssmatrix(B, axis=0, rows=A_mat.shape[0]) # Convert desired poles to numpy array placed_eigs = np.atleast_1d(np.squeeze(np.asarray(p))) result = place_poles(A_mat, B_mat, placed_eigs, method='YT') K = result.gain_matrix return K def place_varga(A, B, p, dtime=False, alpha=None): """Place closed loop eigenvalues using Varga method. K = place_varga(A, B, p, dtime=False, alpha=None) Parameters ---------- A : 2D array_like Dynamics matrix. B : 2D array_like Input matrix. p : 1D array_like Desired eigenvalue locations. dtime : bool, optional False (default) for continuous-time pole placement or True for discrete time. alpha : float, optional If `dtime` is false then place_varga will leave the eigenvalues with real part less than alpha untouched. If `dtime` is true then place_varga will leave eigenvalues with modulus less than alpha untouched. By default (alpha=None), place_varga computes alpha such that all poles will be placed. Returns ------- K : 2D array Gain such that A - B K has eigenvalues given in p. See Also -------- place, place_acker Notes ----- This function is a wrapper for the Slycot function sb01bd, which implements the pole placement algorithm of Varga [1]_. In contrast to the algorithm used by `place`, the Varga algorithm can place multiple poles at the same location. The placement, however, may not be as robust. References ---------- .. [1] Varga A. "A Schur method for pole assignment." IEEE Trans. Automatic Control, Vol. AC-26, pp. 517-519, 1981. Examples -------- >>> A = [[-1, -1], [0, 1]] >>> B = [[0], [1]] >>> K = ct.place_varga(A, B, [-2, -5]) """ # Make sure that Slycot is installed try: from slycot import sb01bd except ImportError: raise ControlSlycot("can't find slycot module sb01bd") # Convert the system inputs to NumPy arrays A_mat = _ssmatrix(A, square=True, name="A") B_mat = _ssmatrix(B, axis=0, rows=A_mat.shape[0]) # Compute the system eigenvalues and convert poles to numpy array system_eigs = np.linalg.eig(A_mat)[0] placed_eigs = np.atleast_1d(np.squeeze(np.asarray(p))) # Need a character parameter for SB01BD if dtime: DICO = 'D' else: DICO = 'C' if alpha is None: # SB01BD ignores eigenvalues with real part less than alpha # (if DICO='C') or with modulus less than alpha # (if DICO = 'D'). if dtime: # For discrete time, Slycot only cares about modulus, so just make # alpha the smallest it can be. alpha = 0.0 else: # Choosing alpha=min_eig is insufficient and can lead to an # error or not having all the eigenvalues placed that we wanted. # Evidently, what python thinks are the eigs is not precisely # the same as what Slycot thinks are the eigs. So we need some # numerical breathing room. The following is pretty heuristic, # but does the trick alpha = -2*abs(min(system_eigs.real)) elif dtime and alpha < 0.0: raise ValueError("Discrete time systems require alpha > 0") # Call Slycot routine to place the eigenvalues A_z, w, nfp, nap, nup, F, Z = \ sb01bd(B_mat.shape[0], B_mat.shape[1], len(placed_eigs), alpha, A_mat, B_mat, placed_eigs, DICO) # Return the gain matrix, with MATLAB gain convention return -F # Contributed by Roberto Bucher def place_acker(A, B, poles): """Pole placement using Ackermann method. Call: K = place_acker(A, B, poles) Parameters ---------- A, B : 2D array_like State and input matrix of the system. poles : 1D array_like Desired eigenvalue locations. Returns ------- K : 2D array Gains such that A - B K has given eigenvalues. See Also -------- place, place_varga """ # Convert the inputs to matrices A = _ssmatrix(A, square=True, name="A") B = _ssmatrix(B, axis=0, rows=A.shape[0], name="B") # Make sure the system is controllable ct = ctrb(A, B) if np.linalg.matrix_rank(ct) != A.shape[0]: raise ValueError("System not reachable; pole placement invalid") # Compute the desired characteristic polynomial p = np.real(np.poly(poles)) # Place the poles using Ackermann's method # TODO: compute pmat using Horner's method (O(n) instead of O(n^2)) n = np.size(p) pmat = p[n-1] * np.linalg.matrix_power(A, 0) for i in np.arange(1, n): pmat = pmat + p[n-i-1] * np.linalg.matrix_power(A, i) K = np.linalg.solve(ct, pmat) K = K[-1, :] # Extract the last row return K def lqr(*args, **kwargs): r"""lqr(A, B, Q, R[, N]) Linear quadratic regulator design. The lqr() function computes the optimal state feedback controller u = -K x that minimizes the quadratic cost .. math:: J = \int_0^\infty (x' Q x + u' R u + 2 x' N u) dt The function can be called with either 3, 4, or 5 arguments: * ``K, S, E = lqr(sys, Q, R)`` * ``K, S, E = lqr(sys, Q, R, N)`` * ``K, S, E = lqr(A, B, Q, R)`` * ``K, S, E = lqr(A, B, Q, R, N)`` where `sys` is an `LTI` object, and `A`, `B`, `Q`, `R`, and `N` are 2D arrays or matrices of appropriate dimension. Parameters ---------- A, B : 2D array_like Dynamics and input matrices. sys : LTI `StateSpace` system Linear system. Q, R : 2D array State and input weight matrices. N : 2D array, optional Cross weight matrix. integral_action : ndarray, optional If this keyword is specified, the controller includes integral action in addition to state feedback. The value of the `integral_action` keyword should be an ndarray that will be multiplied by the current state to generate the error for the internal integrator states of the control law. The number of outputs that are to be integrated must match the number of additional rows and columns in the `Q` matrix. method : str, optional Set the method used for computing the result. Current methods are 'slycot' and 'scipy'. If set to None (default), try 'slycot' first and then 'scipy'. Returns ------- K : 2D array State feedback gains. S : 2D array Solution to Riccati equation. E : 1D array Eigenvalues of the closed loop system. See Also -------- lqe, dlqr, dlqe Notes ----- If the first argument is an LTI object, then this object will be used to define the dynamics and input matrices. Furthermore, if the LTI object corresponds to a discrete-time system, the `dlqr` function will be called. Examples -------- >>> K, S, E = lqr(sys, Q, R, [N]) # doctest: +SKIP >>> K, S, E = lqr(A, B, Q, R, [N]) # doctest: +SKIP """ # # Process the arguments and figure out what inputs we received # # If we were passed a discrete-time system as the first arg, use dlqr() if isinstance(args[0], LTI) and isdtime(args[0], strict=True): # Call dlqr return dlqr(*args, **kwargs) # Get the system description if (len(args) < 3): raise ControlArgument("not enough input arguments") # If we were passed a state space system, use that to get system matrices if isinstance(args[0], StateSpace): A = np.array(args[0].A, ndmin=2, dtype=float) B = np.array(args[0].B, ndmin=2, dtype=float) index = 1 elif isinstance(args[0], LTI): # Don't allow other types of LTI systems raise ControlArgument("LTI system must be in state space form") else: # Arguments should be A and B matrices A = np.array(args[0], ndmin=2, dtype=float) B = np.array(args[1], ndmin=2, dtype=float) index = 2 # Get the weighting matrices (converting to matrices, if needed) Q = np.array(args[index], ndmin=2, dtype=float) R = np.array(args[index+1], ndmin=2, dtype=float) if (len(args) > index + 2): N = np.array(args[index+2], ndmin=2, dtype=float) else: N = None # # Process keywords # # Get the method to use (if specified as a keyword) method = kwargs.pop('method', None) # See if we should augment the controller with integral feedback integral_action = kwargs.pop('integral_action', None) if integral_action is not None: # Figure out the size of the system nstates = A.shape[0] ninputs = B.shape[1] # Make sure that the integral action argument is the right type if not isinstance(integral_action, np.ndarray): raise ControlArgument("Integral action must pass an array") elif integral_action.shape[1] != nstates: raise ControlArgument( "Integral gain size must match system state size") # Process the states to be integrated nintegrators = integral_action.shape[0] C = integral_action # Augment the system with integrators A = np.block([ [A, np.zeros((nstates, nintegrators))], [C, np.zeros((nintegrators, nintegrators))] ]) B = np.vstack([B, np.zeros((nintegrators, ninputs))]) if kwargs: raise TypeError("unrecognized keywords: ", str(kwargs)) # Compute the result (dimension and symmetry checking done in care()) X, L, G = care(A, B, Q, R, N, None, method=method, _Ss="N") return G, X, L def dlqr(*args, **kwargs): r"""dlqr(A, B, Q, R[, N]) Discrete-time linear quadratic regulator design. The dlqr() function computes the optimal state feedback controller u[n] = - K x[n] that minimizes the quadratic cost .. math:: J = \sum_0^\infty (x[n]' Q x[n] + u[n]' R u[n] + 2 x[n]' N u[n]) The function can be called with either 3, 4, or 5 arguments: * ``dlqr(dsys, Q, R)`` * ``dlqr(dsys, Q, R, N)`` * ``dlqr(A, B, Q, R)`` * ``dlqr(A, B, Q, R, N)`` where `dsys` is a discrete-time `StateSpace` system, and `A`, `B`, `Q`, `R`, and `N` are 2d arrays of appropriate dimension (`dsys.dt` must not be 0.) Parameters ---------- A, B : 2D array Dynamics and input matrices. dsys : LTI `StateSpace` Discrete-time linear system. Q, R : 2D array State and input weight matrices. N : 2D array, optional Cross weight matrix. integral_action : ndarray, optional If this keyword is specified, the controller includes integral action in addition to state feedback. The value of the `integral_action` keyword should be an ndarray that will be multiplied by the current state to generate the error for the internal integrator states of the control law. The number of outputs that are to be integrated must match the number of additional rows and columns in the `Q` matrix. method : str, optional Set the method used for computing the result. Current methods are 'slycot' and 'scipy'. If set to None (default), try 'slycot' first and then 'scipy'. Returns ------- K : 2D array State feedback gains. S : 2D array Solution to Riccati equation. E : 1D array Eigenvalues of the closed loop system. See Also -------- lqr, lqe, dlqe Examples -------- >>> K, S, E = dlqr(dsys, Q, R, [N]) # doctest: +SKIP >>> K, S, E = dlqr(A, B, Q, R, [N]) # doctest: +SKIP """ # # Process the arguments and figure out what inputs we received # # Get the system description if (len(args) < 3): raise ControlArgument("not enough input arguments") # If we were passed a continues time system as the first arg, raise error if isinstance(args[0], LTI) and isctime(args[0], strict=True): raise ControlArgument("dsys must be discrete time (dt != 0)") # If we were passed a state space system, use that to get system matrices if isinstance(args[0], StateSpace): A = np.array(args[0].A, ndmin=2, dtype=float) B = np.array(args[0].B, ndmin=2, dtype=float) index = 1 elif isinstance(args[0], LTI): # Don't allow other types of LTI systems raise ControlArgument("LTI system must be in state space form") else: # Arguments should be A and B matrices A = np.array(args[0], ndmin=2, dtype=float) B = np.array(args[1], ndmin=2, dtype=float) index = 2 # Get the weighting matrices (converting to matrices, if needed) Q = np.array(args[index], ndmin=2, dtype=float) R = np.array(args[index+1], ndmin=2, dtype=float) if (len(args) > index + 2): N = np.array(args[index+2], ndmin=2, dtype=float) else: N = np.zeros((Q.shape[0], R.shape[1])) # # Process keywords # # Get the method to use (if specified as a keyword) method = kwargs.pop('method', None) # See if we should augment the controller with integral feedback integral_action = kwargs.pop('integral_action', None) if integral_action is not None: # Figure out the size of the system nstates = A.shape[0] ninputs = B.shape[1] if not isinstance(integral_action, np.ndarray): raise ControlArgument("Integral action must pass an array") elif integral_action.shape[1] != nstates: raise ControlArgument( "Integral gain size must match system state size") else: nintegrators = integral_action.shape[0] C = integral_action # Augment the system with integrators A = np.block([ [A, np.zeros((nstates, nintegrators))], [C, np.eye(nintegrators)] ]) B = np.vstack([B, np.zeros((nintegrators, ninputs))]) if kwargs: raise TypeError("unrecognized keywords: ", str(kwargs)) # Compute the result (dimension and symmetry checking done in dare()) S, E, K = dare(A, B, Q, R, N, method=method, _Ss="N") return K, S, E # Function to create an I/O systems representing a state feedback controller def create_statefbk_iosystem( sys, gain, feedfwd_gain=None, integral_action=None, estimator=None, controller_type=None, xd_labels=None, ud_labels=None, ref_labels=None, feedfwd_pattern='trajgen', gainsched_indices=None, gainsched_method='linear', control_indices=None, state_indices=None, name=None, inputs=None, outputs=None, states=None, params=None, **kwargs): r"""Create an I/O system using a (full) state feedback controller. This function creates an input/output system that implements a state feedback controller of the form .. math:: u = u_d - K_p (x - x_d) - K_i \int(C x - C x_d) by calling ctrl, clsys = ct.create_statefbk_iosystem(sys, K) where `sys` is the process dynamics and `K` is the state (+ integral) feedback gain (e.g., from LQR). The function returns the controller `ctrl` and the closed loop systems `clsys`, both as I/O systems. A gain scheduled controller can also be created, by passing a list of gains and a corresponding list of values of a set of scheduling variables. In this case, the controller has the form .. math:: u = u_d - K_p(\mu) (x - x_d) - K_i(\mu) \int(C x - C x_d) where :math:`\mu` represents the scheduling variable. Alternatively, a controller of the form .. math:: u = k_f r - K_p x - K_i \int(C x - r) can be created by calling ctrl, clsys = ct.create_statefbk_iosystem( sys, K, kf, feedfwd_pattern='refgain') In either form, an estimator can also be used to compute the estimated state from the input and output measurements. Parameters ---------- sys : `NonlinearIOSystem` The I/O system that represents the process dynamics. If no estimator is given, the output of this system should represent the full state. gain : ndarray, tuple, or I/O system If an array is given, it represents the state feedback gain (`K`). This matrix defines the gains to be applied to the system. If `integral_action` is None, then the dimensions of this array should be (sys.ninputs, sys.nstates). If `integral action` is set to a matrix or a function, then additional columns represent the gains of the integral states of the controller. If a tuple is given, then it specifies a gain schedule. The tuple should be of the form ``(gains, points)`` where gains is a list of gains `K_j` and points is a list of values `mu_j` at which the gains are computed. The `gainsched_indices` parameter should be used to specify the scheduling variables. If an I/O system is given, the error e = x - xd is passed to the system and the output is used as the feedback compensation term. feedfwd_gain : array_like, optional Specify the feedforward gain, `k_f`. Used only for the reference gain design pattern. If not given and if `sys` is a `StateSpace` (linear) system, will be computed as -1/(C (A-BK)^{-1}) B. feedfwd_pattern : str, optional If set to 'refgain', the reference gain design pattern is used to create the controller instead of the trajectory generation ('trajgen') pattern. integral_action : ndarray, optional If this keyword is specified, the controller can include integral action in addition to state feedback. The value of the `integral_action` keyword should be an ndarray that will be multiplied by the current and desired state to generate the error for the internal integrator states of the control law. estimator : `NonlinearIOSystem`, optional If an estimator is provided, use the states of the estimator as the system inputs for the controller. gainsched_indices : int, slice, or list of int or str, optional If a gain scheduled controller is specified, specify the indices of the controller input to use for scheduling the gain. The input to the controller is the desired state `x_d`, the desired input `u_d`, and the system state `x` (or state estimate `xhat`, if an estimator is given). If value is an integer `q`, the first `q` values of the ``[x_d, u_d, x]`` vector are used. Otherwise, the value should be a slice or a list of indices. The list of indices can be specified as either integer offsets or as signal names. The default is to use the desired state `x_d`. gainsched_method : str, optional The method to use for gain scheduling. Possible values are 'linear' (default), 'nearest', and 'cubic'. More information is available in `scipy.interpolate.griddata`. For points outside of the convex hull of the scheduling points, the gain at the nearest point is used. controller_type : 'linear' or 'nonlinear', optional Set the type of controller to create. The default for a linear gain is a linear controller implementing the LQR regulator. If the type is 'nonlinear', a `NonlinearIOSystem` is created instead, with the gain `K` as a parameter (allowing modifications of the gain at runtime). If the gain parameter is a tuple, then a nonlinear, gain-scheduled controller is created. Returns ------- ctrl : `NonlinearIOSystem` Input/output system representing the controller. For the 'trajgen' design pattern (default), this system takes as inputs the desired state `x_d`, the desired input `u_d`, and either the system state `x` or the estimated state `xhat`. It outputs the controller action `u` according to the formula u = u_d - K(x - x_d). For the 'refgain' design pattern, the system takes as inputs the reference input `r` and the system or estimated state. If the keyword `integral_action` is specified, then an additional set of integrators is included in the control system (with the gain matrix `K` having the integral gains appended after the state gains). If a gain scheduled controller is specified, the gain (proportional and integral) are evaluated using the scheduling variables specified by `gainsched_indices`. clsys : `NonlinearIOSystem` Input/output system representing the closed loop system. This system takes as inputs the desired trajectory (x_d, u_d) and outputs the system state `x` and the applied input `u` (vertically stacked). Other Parameters ---------------- control_indices : int, slice, or list of int or str, optional Specify the indices of the system inputs that should be determined by the state feedback controller. If value is an integer `m`, the first `m` system inputs are used. Otherwise, the value should be a slice or a list of indices. The list of indices can be specified as either integer offsets or as system input signal names. If not specified, defaults to the system inputs. state_indices : int, slice, or list of int or str, optional Specify the indices of the system (or estimator) outputs that should be used by the state feedback controller. If value is an integer `n`, the first `n` system states are used. Otherwise, the value should be a slice or a list of indices. The list of indices can be specified as either integer offsets or as estimator/system output signal names. If not specified, defaults to the system states. xd_labels, ud_labels, ref_labels : str or list of str, optional Set the name of the signals to use for the desired state and inputs or the reference inputs (for the 'refgain' design pattern). If a single string is specified, it should be a format string using the variable `i` as an index. Otherwise, a list of strings matching the size of x_d and u_d, respectively, should be used. Default is "xd[{i}]" for xd_labels and "ud[{i}]" for ud_labels. These settings can also be overridden using the `inputs` keyword. inputs, outputs, states : str, or list of str, optional List of strings that name the individual signals of the transformed system. If not given, the inputs, outputs, and states are the same as the original system. name : string, optional System name. If unspecified, a generic name 'sys[id]' is generated with a unique integer id. params : dict, optional System parameter values. By default, these will be copied from `sys` and `ctrl`, but can be overridden with this keyword. Examples -------- >>> import control as ct >>> import numpy as np >>> >>> A = [[0, 1], [-0.5, -0.1]] >>> B = [[0], [1]] >>> C = np.eye(2) >>> D = np.zeros((2, 1)) >>> sys = ct.ss(A, B, C, D) >>> >>> Q = np.eye(2) >>> R = np.eye(1) >>> >>> K, _, _ = ct.lqr(sys,Q,R) >>> ctrl, clsys = ct.create_statefbk_iosystem(sys, K) """ # Make sure that we were passed an I/O system as an input if not isinstance(sys, NonlinearIOSystem): raise ControlArgument("Input system must be I/O system") # Process keywords params = sys.params if params is None else params controller_type = _process_legacy_keyword( kwargs, 'type', 'controller_type', controller_type) if kwargs: raise TypeError("unrecognized keywords: ", str(kwargs)) # Check for consistency of positional parameters if feedfwd_gain is not None and feedfwd_pattern != 'refgain': raise ControlArgument( "feedfwd_gain specified but feedfwd_pattern != 'refgain'") # Figure out what inputs to the system to use control_indices = _process_indices( control_indices, 'control', sys.input_labels, sys.ninputs) sys_ninputs = len(control_indices) # Decide what system is going to pass the states to the controller if estimator is None: estimator = sys # Figure out what outputs (states) from the system/estimator to use state_indices = _process_indices( state_indices, 'state', estimator.state_labels, sys.nstates) sys_nstates = len(state_indices) # Make sure the system/estimator states are proper dimension if estimator.noutputs < sys_nstates: # If no estimator, make sure that the system has all states as outputs raise ControlArgument( ("system" if estimator == sys else "estimator") + " output must include the full state") elif estimator == sys: # Issue a warning if we can't verify state output if (isinstance(sys, NonlinearIOSystem) and not isinstance(sys, StateSpace) and sys.outfcn is not None) or \ (isinstance(sys, StateSpace) and not (np.all(sys.C[np.ix_(state_indices, state_indices)] == np.eye(sys_nstates)) and np.all(sys.D[state_indices, :] == 0))): warnings.warn("cannot verify system output is system state") # See whether we should implement integral action nintegrators = 0 if integral_action is not None: if not isinstance(integral_action, np.ndarray): raise ControlArgument("Integral action must pass an array") C = np.atleast_2d(integral_action) if C.shape[1] != sys_nstates: raise ControlArgument( "Integral gain size must match system state size") nintegrators = C.shape[0] else: # Create a C matrix with no outputs, just in case update gets called C = np.zeros((0, sys_nstates)) # Check to make sure that state feedback has the right shape if isinstance(gain, np.ndarray): K = gain if K.shape != (sys_ninputs, estimator.noutputs + nintegrators): raise ControlArgument( f'control gain must be an array of size {sys_ninputs}' f' x {sys_nstates}' + (f'+{nintegrators}' if nintegrators > 0 else '')) gainsched = False elif isinstance(gain, tuple): # Check for gain scheduled controller if len(gain) != 2: raise ControlArgument("gain must be a 2-tuple for gain scheduling") elif feedfwd_pattern != 'trajgen': raise NotImplementedError( "Gain scheduling is not implemented for pattern " f"'{feedfwd_pattern}'") gains, points = gain[0:2] # Stack gains and points if past as a list gains = np.stack(gains) points = np.stack(points) gainsched = True elif isinstance(gain, NonlinearIOSystem) and feedfwd_pattern != 'refgain': if controller_type not in ['iosystem', None]: raise ControlArgument( f"incompatible controller type '{controller_type}'") fbkctrl = gain controller_type = 'iosystem' gainsched = False else: raise ControlArgument("gain must be an array or a tuple") # Decide on the type of system to create if gainsched and controller_type == 'linear': raise ControlArgument( "controller_type 'linear' not allowed for" " gain scheduled controller") elif controller_type is None: controller_type = 'nonlinear' if gainsched else 'linear' elif controller_type not in {'linear', 'nonlinear', 'iosystem'}: raise ControlArgument(f"unknown controller_type '{controller_type}'") # Figure out the labels to use if feedfwd_pattern == 'trajgen': xd_labels = _process_labels(xd_labels, 'xd', [ 'xd[{i}]'.format(i=i) for i in range(sys_nstates)]) ud_labels = _process_labels(ud_labels, 'ud', [ 'ud[{i}]'.format(i=i) for i in range(sys_ninputs)]) # Create the signal and system names if inputs is None: inputs = xd_labels + ud_labels + estimator.output_labels elif feedfwd_pattern == 'refgain': ref_labels = _process_labels(ref_labels, 'r', [ f'r[{i}]' for i in range(sys_ninputs)]) if inputs is None: inputs = ref_labels + estimator.output_labels else: raise NotImplementedError(f"unknown pattern '{feedfwd_pattern}'") if outputs is None: outputs = [sys.input_labels[i] for i in control_indices] if states is None: states = nintegrators # Process gain scheduling variables, if present if gainsched: # Create a copy of the scheduling variable indices (default = xd) gainsched_indices = _process_indices( gainsched_indices, 'gainsched', inputs, sys_nstates) # If points is a 1D list, convert to 2D if points.ndim == 1: points = points.reshape(-1, 1) # Make sure the scheduling variable indices are the right length if len(gainsched_indices) != points.shape[1]: raise ControlArgument( "length of gainsched_indices must match dimension of" " scheduling variables") # Create interpolating function if points.shape[1] < 2: _interp = sp.interpolate.interp1d( points[:, 0], gains, axis=0, kind=gainsched_method) _nearest = sp.interpolate.interp1d( points[:, 0], gains, axis=0, kind='nearest') elif gainsched_method == 'nearest': _interp = sp.interpolate.NearestNDInterpolator(points, gains) def _nearest(mu): raise SystemError(f"could not find nearest gain at mu = {mu}") elif gainsched_method == 'linear': _interp = sp.interpolate.LinearNDInterpolator(points, gains) _nearest = sp.interpolate.NearestNDInterpolator(points, gains) elif gainsched_method == 'cubic': _interp = sp.interpolate.CloughTocher2DInterpolator(points, gains) _nearest = sp.interpolate.NearestNDInterpolator(points, gains) else: raise ControlArgument( f"unknown gain scheduling method '{gainsched_method}'") def _compute_gain(mu): K = _interp(mu) if np.isnan(K).any(): K = _nearest(mu) return K # Define the controller system if controller_type == 'nonlinear' and feedfwd_pattern == 'trajgen': # Create an I/O system for the state feedback gains def _control_update(t, states, inputs, params): # Split input into desired state, nominal input, and current state xd_vec = inputs[0:sys_nstates] x_vec = inputs[-sys_nstates:] # Compute the integral error in the xy coordinates return C @ (x_vec - xd_vec) def _control_output(t, states, inputs, params): if gainsched: mu = inputs[gainsched_indices] K_ = _compute_gain(mu) else: K_ = params.get('K') # Split input into desired state, nominal input, and current state xd_vec = inputs[0:sys_nstates] ud_vec = inputs[sys_nstates:sys_nstates + sys_ninputs] x_vec = inputs[-sys_nstates:] # Compute the control law u = ud_vec - K_[:, 0:sys_nstates] @ (x_vec - xd_vec) if nintegrators > 0: u -= K_[:, sys_nstates:] @ states return u ctrl_params = {} if gainsched else {'K': K} ctrl = NonlinearIOSystem( _control_update, _control_output, name=name, inputs=inputs, outputs=outputs, states=states, params=ctrl_params) elif controller_type == 'iosystem' and feedfwd_pattern == 'trajgen': # Use the passed system to compute feedback compensation def _control_update(t, states, inputs, params): # Split input into desired state, nominal input, and current state xd_vec = inputs[0:sys_nstates] x_vec = inputs[-sys_nstates:] # Compute the integral error in the xy coordinates return fbkctrl.updfcn(t, states, (x_vec - xd_vec), params) def _control_output(t, states, inputs, params): # Split input into desired state, nominal input, and current state xd_vec = inputs[0:sys_nstates] ud_vec = inputs[sys_nstates:sys_nstates + sys_ninputs] x_vec = inputs[-sys_nstates:] # Compute the control law return ud_vec + fbkctrl.outfcn(t, states, (x_vec - xd_vec), params) # TODO: add a way to pass parameters ctrl = NonlinearIOSystem( _control_update, _control_output, name=name, inputs=inputs, outputs=outputs, states=fbkctrl.state_labels, dt=fbkctrl.dt) elif controller_type in 'linear' and feedfwd_pattern == 'trajgen': # Create the matrices implementing the controller if isctime(sys): # Continuous time: integrator A_lqr = np.zeros((C.shape[0], C.shape[0])) else: # Discrete time: summer A_lqr = np.eye(C.shape[0]) B_lqr = np.hstack([-C, np.zeros((C.shape[0], sys_ninputs)), C]) C_lqr = -K[:, sys_nstates:] D_lqr = np.hstack([ K[:, 0:sys_nstates], np.eye(sys_ninputs), -K[:, 0:sys_nstates] ]) ctrl = ss( A_lqr, B_lqr, C_lqr, D_lqr, dt=sys.dt, name=name, inputs=inputs, outputs=outputs, states=states) elif feedfwd_pattern == 'refgain': if controller_type not in ['linear', 'iosystem']: raise ControlArgument( "refgain design pattern only supports linear controllers") if feedfwd_gain is None: raise ControlArgument( "'feedfwd_gain' required for reference gain pattern") # Check to make sure the reference gain is valid Kf = np.atleast_2d(feedfwd_gain) if Kf.ndim != 2 or Kf.shape[0] != sys.ninputs or \ Kf.shape[1] != sys.ninputs: raise ControlArgument("feedfwd_gain is not the right shape") # Create the matrices implementing the controller # [r, x]->[u]: u = k_f r - K_p x - K_i \int(C x - r) if isctime(sys): # Continuous time: integrator A_lqr = np.zeros((C.shape[0], C.shape[0])) else: # Discrete time: summer A_lqr = np.eye(C.shape[0]) B_lqr = np.hstack([-np.eye(C.shape[0], sys_ninputs), C]) C_lqr = -K[:, sys_nstates:] # integral gain (opt) D_lqr = np.hstack([Kf, -K]) ctrl = ss( A_lqr, B_lqr, C_lqr, D_lqr, dt=sys.dt, name=name, inputs=inputs, outputs=outputs, states=states) else: raise ControlArgument(f"unknown controller_type '{controller_type}'") # Define the closed loop system inplist=inputs[:-sys.nstates] input_labels=inputs[:-sys.nstates] outlist=sys.output_labels + [sys.input_labels[i] for i in control_indices] output_labels=sys.output_labels + \ [sys.input_labels[i] for i in control_indices] closed = interconnect( [sys, ctrl] if estimator == sys else [sys, ctrl, estimator], name=sys.name + "_" + ctrl.name, add_unused=True, inplist=inplist, inputs=input_labels, outlist=outlist, outputs=output_labels, params= ctrl.params | params ) return ctrl, closed def ctrb(A, B, t=None): """Controllability matrix. Parameters ---------- A, B : array_like or string Dynamics and input matrix of the system. t : None or integer Maximum time horizon of the controllability matrix, max = A.shape[0]. Returns ------- C : 2D array Controllability matrix. Examples -------- >>> G = ct.tf2ss([1], [1, 2, 3]) >>> C = ct.ctrb(G.A, G.B) >>> np.linalg.matrix_rank(C) np.int64(2) """ # Convert input parameters to matrices (if they aren't already) A = _ssmatrix(A, square=True, name="A") n = A.shape[0] B = _ssmatrix(B, axis=0, rows=n, name="B") m = B.shape[1] if t is None or t > n: t = n # Construct the controllability matrix ctrb = np.zeros((n, t * m)) ctrb[:, :m] = B for k in range(1, t): ctrb[:, k * m:(k + 1) * m] = np.dot(A, ctrb[:, (k - 1) * m:k * m]) return ctrb def obsv(A, C, t=None): """Observability matrix. Parameters ---------- A, C : array_like or string Dynamics and output matrix of the system. t : None or integer Maximum time horizon of the controllability matrix, max = A.shape[0]. Returns ------- O : 2D array Observability matrix. Examples -------- >>> G = ct.tf2ss([1], [1, 2, 3]) >>> C = ct.obsv(G.A, G.C) >>> np.linalg.matrix_rank(C) np.int64(2) """ # Convert input parameters to matrices (if they aren't already) A = _ssmatrix(A, square=True, name="A") n = A.shape[0] C = _ssmatrix(C, cols=n, name="C") p = C.shape[0] if t is None or t > n: t = n # Construct the observability matrix obsv = np.zeros((t * p, n)) obsv[:p, :] = C for k in range(1, t): obsv[k * p:(k + 1) * p, :] = np.dot(obsv[(k - 1) * p:k * p, :], A) return obsv def gram(sys, type): """Gramian (controllability or observability). Parameters ---------- sys : `StateSpace` System description. type : String Type of desired computation. `type` is either 'c' (controllability) or 'o' (observability). To compute the Cholesky factors of Gramians use 'cf' (controllability) or 'of' (observability). Returns ------- gram : 2D array Gramian of system. Raises ------ ValueError * If system is not instance of `StateSpace` class, or * if `type` is not 'c', 'o', 'cf' or 'of', or * if system is unstable (sys.A has eigenvalues not in left half plane). ControlSlycot If slycot routine sb03md cannot be found or if slycot routine sb03od cannot be found. Examples -------- >>> G = ct.rss(4) >>> Wc = ct.gram(G, 'c') >>> Wo = ct.gram(G, 'o') >>> Rc = ct.gram(G, 'cf') # where Wc = Rc' * Rc >>> Ro = ct.gram(G, 'of') # where Wo = Ro' * Ro """ # Check for ss system object if not isinstance(sys, statesp.StateSpace): raise ValueError("System must be StateSpace!") if type not in ['c', 'o', 'cf', 'of']: raise ValueError("That type is not supported!") # Check if system is continuous or discrete if sys.isctime(): dico = 'C' # TODO: Check system is stable, perhaps a utility in ctrlutil.py # or a method of the StateSpace class? if np.any(np.linalg.eigvals(sys.A).real >= 0.0): raise ValueError("Oops, the system is unstable!") else: assert sys.isdtime() dico = 'D' if np.any(np.abs(sys.poles()) >= 1.): raise ValueError("Oops, the system is unstable!") if type == 'c' or type == 'o': # Compute Gramian by the Slycot routine sb03md # make sure Slycot is installed if sb03md is None: raise ControlSlycot("can't find slycot module sb03md") if type == 'c': tra = 'T' C = -sys.B @ sys.B.T elif type == 'o': tra = 'N' C = -sys.C.T @ sys.C n = sys.nstates U = np.zeros((n, n)) A = np.array(sys.A) # convert to NumPy array for slycot X, scale, sep, ferr, w = sb03md( n, C, A, U, dico, job='X', fact='N', trana=tra) gram = X return gram elif type == 'cf' or type == 'of': # Compute Cholesky factored Gramian from Slycot routine sb03od if sb03od is None: raise ControlSlycot("can't find slycot module sb03od") tra = 'N' n = sys.nstates Q = np.zeros((n, n)) A = np.array(sys.A) # convert to NumPy array for slycot if type == 'cf': m = sys.B.shape[1] B = np.zeros_like(A) B[0:m, 0:n] = sys.B.transpose() X, scale, w = sb03od( n, m, A.transpose(), Q, B, dico, fact='N', trans=tra) elif type == 'of': m = sys.C.shape[0] C = np.zeros_like(A) C[0:n, 0:m] = sys.C.transpose() X, scale, w = sb03od( n, m, A, Q, C.transpose(), dico, fact='N', trans=tra) gram = X return gram # Short versions of functions acker = place_acker